Theorem elabs 1966

Description: Membership in a class abstraction, expressed in terms of class substitution.

Hypothesis

Ref	Expression
elabs.1	|- A e. V

Assertion

Ref	Expression
elabs	|- (A e. {x | ph} <-> [A / x]ph)

Proof of Theorem elabs

Step	Hyp	Ref	Expression
1		elabs2 1964	. 2 |- (A e. {x | ph} <-> (A e. V /\ [A / x]ph))
2		elabs.1	. 2 |- A e. V
3	1, 2	mpbiran 728	1 |- (A e. {x | ph} <-> [A / x]ph)

Syntax hints:   <-> wb 146   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811

This theorem is referenced by:  intab 2560  hta 4728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-sbc 1942

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